The origin of the cold gas in central galaxies in groups is still a matter of debate. We present Multi-Unit Spectroscopic Explorer (MUSE) observations of 18 optically selected local (z ≤0.017) Brightest Group Galaxies (BGGs) to study the kinematics and distribution of the optical emission-line gas. MUSE observations reveal a distribution of gas morphologies including ten complex networks of filaments extending up to ∼10 kpc to two compact (<3 kpc) and five extended (>5 kpc) disk-dominated structures. Some rotating disks show rings and elongated structures arising from the central disk. The kinematics of the stellar component is mainly rotationdominated, which is very different from the disturbed kinematics and distribution found in the filamentary sources. The ionized gas is kinematically decoupled from the stellar component for most systems, suggesting an external origin for the gas. We find also that the Hα luminosity correlates with the cold molecular gas mass. By exploring the thermodynamical properties of the X-ray atmospheres, we find that the filamentary structures and compact disks are found in systems with small central entropy values, K, and t cool /t eddy ratios. This suggests that, like for Brightest Cluster Galaxies (BCGs) in cool core clusters, the ionized filaments and the cold gas associated are likely formed from hot halo gas condensations via thermal instabilities, consistently with the Chaotic Cold Accretion simulations (as shown via the C-ratio, Ta t , and k-plot). We note that the presence of gaseous rotating disks is more frequent than in BCGs. An explanation for the origin of the gas in those objects is a contribution to gas fueling by wet-mergers or group satellites, as qualitatively hinted by some sources of the present sample. Nonetheless, we discuss the possibility that some extended disks could also be a transition stage in an evolutionary sequence including filaments, extended disks and compact disks, as described by hot gas condensation models of cooling flows.